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1.3.6 The Universal Property of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$

We now give an alternative description of the homotopy category of an $\infty $-category.

Construction 1.3.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. For $0 \leq i \leq n$, let $C_{i}$ denote the object of $\operatorname{\mathcal{C}}$ given by the image of the $i$th vertex of $\Delta ^ n$. For $0 \leq i \leq j \leq n$, let $f_{ij}: C_ i \rightarrow C_ j$ denote the image under $\sigma $ of the edge of $\Delta ^ n$ joining the $i$th vertex to the $j$th vertex, and let $[f_{ij}] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( C_ i, C_ j )$ denote the homotopy class of $f_{ij}$. Then we can regard $( \{ C_ i \} _{0 \leq i \leq n}, \{ [f_{ij} ] \} _{0 \leq i \leq j \leq n} )$ as a functor from the linearly ordered set $[n]$ to the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Let $u(\sigma )$ denote the corresponding $n$-simplex of $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$. Then the construction $\sigma \mapsto u(\sigma )$ determines a map of simplicial sets

\[ u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ). \]

The comparison map of Construction 1.3.6.1 has the following universal property:

Proposition 1.3.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ be as in Construction 1.3.6.1. For every category $\operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{ \operatorname{Cat}}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \xrightarrow { \circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \]

is bijective.

Proof. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ be a map of simplicial sets. Then $F$ induces a functor of homotopy categories $G: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{ \operatorname{N}}_{\bullet }(\operatorname{\mathcal{D}}) \simeq \operatorname{\mathcal{D}}$ (where the second identification comes from Example 1.3.5.4). By construction, the map of simplicial sets

\[ \operatorname{\mathcal{C}}\xrightarrow {u} \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \xrightarrow { \operatorname{N}_{\bullet }(G) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]

agrees with $F$ on the vertices and edges of $\operatorname{\mathcal{C}}$, and therefore coincides with $F$ (since a simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is determined by its $1$-dimensional facets; see Remark 1.2.1.3). We leave it to the reader to verify that $G$ is the unique functor with this property. $\square$

Using Proposition 1.3.6.2, we can extend the notion of homotopy category to more general simplicial sets.

Definition 1.3.6.3. Let $\operatorname{\mathcal{C}}$ be a category. We will say that a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ exhibits $\operatorname{\mathcal{C}}$ as the homotopy category of $S_{\bullet }$ if, for every category $\operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \]

is bijective (note that the map on the left is always bijective, by virtue of Proposition 1.2.2.1).

Notation 1.3.6.4. Let $S_{\bullet }$ be a simplicial set. It follows immediately from the definition that if there exists a category $\operatorname{\mathcal{C}}$ and a map $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S_{\bullet }$, then the category $\operatorname{\mathcal{C}}$ is unique up to isomorphism and depends functorially on $S_{\bullet }$. To emphasize this dependence, we will refer to $\operatorname{\mathcal{C}}$ as the homotopy category of $S_{\bullet }$ and denote it by $\mathrm{h} \mathit{S}_{\bullet }$.

Example 1.3.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ constructed in Definition 1.3.5.3 is also a homotopy category of $\operatorname{\mathcal{C}}$ in the sense of Definition 1.3.6.3. More precisely, the map $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ of Construction 1.3.6.1 exhibits $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a homotopy category of $\operatorname{\mathcal{C}}$, by virtue of Proposition 1.3.6.2.

Proposition 1.3.6.6. Let $S_{\bullet }$ be a simplicial set. Then there exists a category $\operatorname{\mathcal{C}}$ and a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a homotopy category of $S_{\bullet }$.

Proof. Let $Q^{\bullet }$ denote the cosimplicial object of $\operatorname{Cat}$ given by the inclusion $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}$. Unwinding the definitions, we see that a homotopy category of $S_{\bullet }$ can be identified with a realization $| S_{\bullet } |^{Q}$, whose existence is a special case of Proposition 1.1.6.18. Alternatively, we can give a direct construction of the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$:

  • The objects of $\mathrm{h} \mathit{S}_{\bullet }$ are the vertices of $S_{\bullet }$.

  • Every edge $e$ of $S_{\bullet }$ determines a morphism $[e]$ in $\mathrm{h} \mathit{S}_{\bullet }$, whose domain is the vertex $d_1(e)$ and whose codomain is the vertex $d_0(e)$.

  • The collection of morphisms in $\mathrm{h} \mathit{S}_{\bullet }$ is generated under composition by morphisms of the form $[e]$, subject only to the relations

    \[ [ s_0(x) ] = \operatorname{id}_ x \text{ for $x \in S_0$ } \quad \quad [ d_1(\sigma ) ] = [ d_0(\sigma ) ] \circ [ d_2(\sigma ) ] \text{ for $\sigma \in S_2$. } \]
$\square$

Corollary 1.3.6.7. The nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint, given on objects by the construction $S_{\bullet } \mapsto \mathrm{h} \mathit{S}_{\bullet }$.

Example 1.3.6.8. Let $G$ be a directed graph (Definition 1.1.4.1) and let $S_{\bullet }$ denote the associated simplicial set of dimension $\leq 1$ (Proposition 1.1.4.9). Then the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ can be described explicitly as follows:

  • The objects of $\mathrm{h} \mathit{S}_{\bullet }$ are the vertices of the graph $G$.

  • Given a pair of vertices $v,w \in \operatorname{Vert}(G)$, a morphism from $v$ to $w$ in $\mathrm{h} \mathit{S}_{\bullet }$ is given by a path from $v$ to $w$ in the directed graph $G$: that is, an ordered sequence of edges $(e_1, e_2, \ldots , e_ n)$ satisfying $s( e_1 ) = v$, $t(e_ n) = w$, and $t( e_ i ) = s( e_{i+1} )$ for $0 < i < n$. Here $s,t: \operatorname{Vert}(G) \rightarrow \operatorname{Edge}(G)$ denote the source and target maps. Moreover, we allow $n=0$ in the case $v = w$ (the empty sequence is regarded as the identity morphism from the vertex $v$ to itself).

  • Composition of morphisms in $\mathrm{h} \mathit{S}_{\bullet }$ is given by concatenation of paths. More precisely, given morphisms $f = (e_1, e_2, \ldots , e_ m)$ from $u$ to $v$ and $g = (e'_1, e'_2, \ldots , e'_ n)$ from $v$ to $w$, the composition $g \circ f$ is given by the sequence $(e_1, e_2, \ldots , e_ m, e'_1, \ldots , e'_ n)$.